1. Show that the function f : R* R* defined by f (x) = 1 x is one-one and onto, where R * is the set of all non-zero real numbers. Is the result true, if the domain R * is replaced by N with co-domain being same as R *? 2. Check the injectivity and surjectivity of the following functions: (i) f : N N given by f (x) = x2 (ii) f : Z Z given by f (x) = x2 (iii) f : R R given by f (x) = x2 (iv) f : N N given by f (x) = x3 (v) f : Z Z given by f (x) = x3 3. Prove that the Greatest Integer Function f : R R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. 4. Show that the Modulus Function f : R R, given by f (x) = | x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x | is x, if x is negative. 5.
Question 1:Show that the function f : R* → R* defined by f (x) = 1 x is one-one and onto, where R * is the set of all non-zero real numbers. Is the result true, if the domain R * is replaced by N with co-domain being same as R *?
Question 2:Check the injectivity and surjectivity of the following functions: (i) f : N → N given by f (x) = x2 (ii) f : Z → Z given by f (x) = x2 (iii) f : R → R given by f (x) = x2 (iv) f : N → N given by f (x) = x3 (v) f : Z → Z given by f (x) = x3
Question 3:Prove that the Greatest Integer Function f : R → R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Question 4:Show that the Modulus Function f : R → R, given by f (x) = | x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x | is – x, if x is negative.
Question 5:Show that the Signum Function f : R → R, given by f x x x x ( ) , , , = > = < ì í ï î ï 1 0 0 0 1 0 if if if is neither one-one nor onto.
Question 6:Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.
Question 7:In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) f : R → R defined by f (x) = 3 – 4x (ii) f : R → R defined by f (x) = 1 + x2
Question 8:Let A and B be sets. Show that f : A × B → B × A such that f (a, b) = (b, a) is bijective function.
Question 9:Let f : N → N be defined by f (n) = n n n n ì + í ïï î ïï 1 2 2 , , if is odd if is even for all n Î N. State whether the function f is bijective. Justify your answer.
Question 10:Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by f (x) = 2 3 x x − − . Is f one-one and onto? Justify your answer.
Question 11:Let f : R → R be defined as f(x) = x4. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.
Question 12:Let f : R → R be defined as f (x) = 3x. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.
